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Wave packets in graphene whose central wave vector is at Dirac points areinvestigated by numerical calculations. Starting from an initial Gaussianfunction, these wave packets form into annular peaks that propagate to alldirections like ripple-rings on water surface. At the beginning, electronicprobability alternates between the central peak and the ripple-rings andtransient oscillation occurs at the center. As time increases, the ripple-ringspropagate at the fixed Fermi speed, and their widths remain unchanged. Theaxial symmetry of the energy dispersion leads to the circular symmetry of thewave packets. The fixed speed and widths, however, are attributed to thelinearity of the energy dispersion. Interference between states thatrespectively belong to two branches of the energy dispersion leads to multipleripple-rings and the probability-density oscillation. In a magnetic field,annular wave packets become confined and no longer propagate to infinity. Ifthe initial Gaussian width differs greatly from the magnetic length, expandingand shrinking ripple-rings form and disappear alternatively in a limitedspread, and the wave packet resumes the Gaussian form frequently. Theprobability thus oscillates persistently between the central peak and theripple-rings. If the initial Gaussian width is close to the magnetic length,the wave packet retains the Gaussian form and its height and width oscillatewith a period determined by the first Landau energy. The wave-packet evolutionis determined jointly by the initial state and the magnetic field, through theelectronic structure of graphene in a magnetic field.

Annular wave packets at Dirac points and probability oscillation in graphene